\documentclass[a4paper]{jpconf}
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\begin{document}
\title{Possibilities of $\mu$SR-technique for study of magnetization
processes in nanocrystal films of ferromagnetic metals}

\author{Yu. M. Belousov}

\address{Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Institutsky lane 9, Russia}

\ead{theorphys@phystech.edu}

\begin{abstract}
It is shown that $\mu$SR-technique allows to study appearance of
ferromagnetic ordering in nanocrystal films of ferromagnetic
metals -- so called ``scale'' phase transition. A relationship of
a microscopic (local) field with macroscopic characteristics just
as an external magnetic field, average magnetization and
saturation magnetization is determined in a model of the
nanocrystal film consisting of crystallographically ordered grains
separated by disordered areas. Expressions for behaviour of a muon
spin polarization ensemble in this kind of structures are obtained
in cases of fast diffusing and nondiffusing muons. It is shown
that experiments with ``slow'' positively charged muons allow to
measure all parameters of this kind of structures and obtain
important information for the study of phase transition physics.
\end{abstract}

\section{Introduction}

Magnetic nanocrystalline metals are of interest for
$\mu$SR-technique since using positive muons and neutron
difraction possess real possibilities to study their bulk
properties\footnote{Nanocrystalline metals are conventionally
accepted to be polycrystals with grain sizes of 10--400\AA.}.
Possible applications of $\mu$SR-technique for study
nanocrystalline ferromagnetic materials were still mentioned at
2000-th \cite{BS}. Nevertheless, no serious experimental
researches were carried out yet. There was emphasized that at
least two problems in the fundamental physics of magnets could be
solved for nanostructured ferromagnets. One of them is the problem
of specific ``scale'' phase transitions, when an existence of a
spontaneous magnetization depends both on temperature and sizes of
a crystal. The mechanism of this size-induced phase transition is
being actively studied (see, e.g. \cite{Tepl} - \cite{Frol}).
Currently, materials with grain sizes too small to exhibit
ferromagnet properties are called superparamagnets. The second
problem is related with the structure and magnetic properties of
domains in nanostructures. They should be strongly differ from the
respective characteristics of ``ordinary'' polycrystalline
ferromagnets. Really, both theory and experiments show that a
microcrystalline grain ($10^2 - 4\cdot 10^2$ \AA) should be a
single domain. The characteristic thickness of domain walls in a
bulk sample\footnote{The characteristic thikness of domain walls
depends on the ratio of exchange and magnetocrystalline anisotropy
energy scale. The presented values are correct for ferromagnets
under consideration.} is $d \sim 10 - 30$ \AA, which is of the
same order of magnitude as the thickness of the intercrystalline
amorphous interface. Thus, the concept of magnetic domains and
domain walls in nanocrystals differs from the observed for
macroscopic polycrystals. In our case we need to consider
magnetized regions without domain grains in a nonmagnetic medium.
Hence, the macroscopic field $B_{\rm dom}$ inside a
crystallographically ordered grain and its dependence on the
external magnetic field $\cal B$ should differ significantly from
the properties of macroscopic samples.

Opportunities of the $\mu$SR-technique for studying ferromagnetic
nanostructured thin films are represented in this report.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Behaviour of muon spin polarization}

Let consider a nanostructured film consists of
crystallographically ordered grains separated by disordered
regions. Spontaneous magnetization can arise only in ordered
regions. Therefore, the local field on a muon depends on wether
the muon stops in a grain interstice or in an intergrain region.
Let consider a situation when a spontaneous magnetization does not
equal to zero. Thus the spin polarization of an ensemble of muons
can be written as the sum
\begin{equation}\label{n1}
  {\bf{\cal P}}(t)= {\bf P}_{\rm cr}(t)+
  {\bf P}_{\rm nc}(t),
\end{equation}
where ${\bf P}_{\rm cr}(t)$ and ${\bf P}_{\rm nc}(t)$ are the
polarization of the fraction of muons stopped in grains and in
intergrain regions, respectively.

For a non-diffusing muons, the polarization is given by well-known
expression (see e.g. \cite{SB})
\begin{equation}\label{n2}
P_i(t) = \mu_{ik}(t)P_k(0)
\end{equation}
where the tensor $\mu_{ik}(t)$ is
\begin{equation}\label{n3}
  \mu_{ik}(t) = n_in_k + (\delta_{ik} - n_in_k)\cos\gamma_{\mu} b_{\mu}t +
   e_{ikl}n_l\sin\gamma_{\mu} b_{\mu}t
\end{equation}
Here ${\bf b}_{\mu}$ is the local field exerted on the muon,
$\gamma_{\mu} = 13.554$ kHz/G is the gyromagnetic ratio and ${\bf
n} = {\bf b}_{\mu}/|{\bf b}_{\mu}|$ is the unit vector along the
magnetic field.

The local field acting on a muon in a grain and an intergrain
region differ significantly. We will be interesting in at first
the field in a crystallographically ordered grain. In general, by
separating a Lorentz sphere around the muon position, we can write
\cite{{SB},{Bel}}:
\begin{equation}\label{n4}
  {\bf b}_{\mu} = {\bf B} - \frac{8\pi}{3}{\bf M} +
  {\bf b}_{\rm dip} + {\bf B}_{\rm cont},
\end{equation}
where ${\bf B}_{\rm cont}$ is the contact field induced by
electrons and ${\bf b}_{\rm dip}$ is the microscopic field induced
by the oriented magnetic dipoles inside the Lorentz sphere. The
contact field can always be written as $  B_{i\,\rm cont}=
K_{ik}B_k$. In cubic crystals, we can set $K_{ik}=\delta_{ik}K$.
Therefore, the contact field causes only an isotropic Knight
shift\footnote{For the simplicity, we can omit the Knight shift in
what follows, although it can make an appreciable contribution in
some cases.}.

In an intergrain region, the spontaneous magnetization is equal to
zero (${\bf M}_{\rm nc}=0$), hence, the dipole fields  ${\bf
b}_{\rm dip}$ could be induced only by the disordered nuclear
magnetic moments. These fields cause inhomogeneous line
broadening, which can be adequately described by a Gaussian in the
case of nondiffusing muons and a simple exponent for rapidly
diffusing muons (see e.g.,\cite{{SB},{A}}). Thus, the behaviour of
the muon spin polarization for integrain regions is controlled by
the local magnetic field ${\bf b}_{\mu}=\langle {\bf B}\rangle
+\delta{\bf b}$, where $\delta {\bf b}$ is the static field
inhomogeneity. The characteristic scale of the field inhomogeneity
in the intergrain region is determined by the magnetization of
grains and the distance between them. Thus, the polarization
precession frequency of muons in the intergrain fraction allows to
determine the average magnetic field in the film. In the case of
nondiffusing muons a Gaussian exponent allows to determine the
characteristic scale of the field inhomogeneity, $\sigma \sim
\gamma_{\mu}\langle \delta{\bf b}^2\rangle/|\langle {\bf
B}\rangle|$. In the case of diffusing muons depolarization rate
depends on a diffusing rapid $\lambda$ and in the fast diffusing
limit becomes negligibly small. It is known that muon can diffuse
rapidly in polycrystal samples (see e.g. \cite{SB} Ch.5 and
references therein). But an irregular structure of an amorphous
intergrain region differs from a structure of polycrystal samples.
So, we can assume that a diffusion of a muon in intergrain regions
is improbable.

If a crystallographically ordered grain has a nonzero spontaneous
magnetization, the microscopic dipole field is induced by the
ordered electron magnetic moments. In this case, inside a Lorentz
sphere, this field can always be written as \cite{{SB},{Bel}}
\begin{equation}\label{n5}
  b_{i\,\rm dip}= -\frac{4\pi}{3}M_i +a_{ik}M_k,
\end{equation}
where the tensor $a_{ik}$ depends on the type of interstice at the
center of which the field is determined. Calculations showed (see
e.g.,\cite{{SB},{Bel}}) that $a_{ik}=\delta_{ik}4\pi/3$ in an fcc
Ni lattice; hence, the microscopic dipole field is zero, ${\bf
b}_{\rm dip}(fcc) =0$.

In an hcp Co lattice, the dipole field is also weak but is nonzero
and has different values in crystallographically nonequivalent
interstices. If we direct the $z$ axis along the hexagonal axis,
we have
\begin{equation}\label{n6}
  \begin{array}{rl}
    \delta a^h_{xx} & = \delta a^h_{yy}=\Delta/2, \quad
    \delta a^h_{zz} = -\Delta \quad \mbox{in octahedral interstitice},\\
    \delta a^t_{xx} &= \delta a^t_{yy}=-\Delta, \quad
    \delta a^t_{zz}=2\Delta \quad \mbox{in tetrahedral interstice},
  \end{array}
\end{equation}
where $\Delta \approx 0.1$. So, the dipole field can be written as
${\bf b}_{\rm dip}(hcc)=\delta a_{ik}M_k$. Therefore, we can
decide the dipole field $b_{\rm dip}\ll M$.

The more complicated picture is observed in a bcc Fe lattice,
where the dipole field is large and depends not only on the
interstice type but also on the direction of the magnetization
vector ${\bf M}$. The components of the tensor $a_{ik}(bcc)$ in
the principal axes are \cite{{SB},{Bel}}
\begin{equation}\label{n7}
  \begin{array}{rl}
     a^h_{xx} & =  a^h_{yy}=-1.165, \quad
     a^h_{zz} = 14.9 \quad \mbox{in octahedral interstice},\\
     a^t_{xx} &= a^t_{yy}=\quad 5.707, \quad
     a^t_{zz}= 1.152 \quad \mbox{in tetrahedral interstice}.
  \end{array}
\end{equation}
Hence, the local field on a muon in a bcc lattice is given by
\begin{equation}\label{n8}
  b_{\mu \, i}(bcc) =  B_i - 4\pi M_i +a_{ik}(bcc)M_k.
\end{equation}

The local field acting on a muon depends on the direction of the
magnetization vector in the grain (see Eq. (\ref{n5})), the
polarization fraction of muons that stop in the
crystallographically ordered grains should be defined by averaging
over all possible orientations of the principal crystallographic
axes.

Let us represent the local field acting on a muon as the sum of
the two components parallel and perpendicular to the film plane
${\bf b}= {\bf b}_{\|}+ {\bf b}_{\perp}$. Than, the polarization
components are defined as
\begin{equation}\label{n9}
  {\cal P}_{\perp}= \langle \frac{b_{\|}}{b}
  {\e}^{{\i}\omega t}\rangle, \quad
  {\cal P}_{\|}= \langle \frac{b_{\perp}}{b}
  \sin\omega t\rangle, \quad\mbox{где} \quad
   \omega= \gamma_{\mu} b=
  \gamma_{\mu}\sqrt{{\bf b}_{\|}^2+ {\bf b}_{\perp}^2}
\end{equation}
The preexponential factors (direction cosines) depend on whether
the external field is perpendicular or parallel to the film plane.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Hierarchy of fields}

To determine the local field acting on a muon implanted into a
target, we need to find the microscopic field in the sample. At
first, the hierarchy of fields in films of nanostructured
ferromagnetic metals should be refined. As for a multidomain
ferromagnet, in addition to macroscopic fields in {\it each} grain
${\bf M}, \, {\bf B}, \, {\bf H}$ and the external with the
respect to {\it entire} sample $\boldsymbol {\cal B}$  we need to
introduce average macroscopic fields in the sample. These fields
are the results of an averaging of all mentioned above macroscopic
fields and the magnetization over all grains and intergrain
regions, $\langle {\bf B}\rangle, \, \langle {\bf H}\rangle$ и
$\langle {\bf M}\rangle$. The last parameters are determined in
macroscopic experiments. The same parameters determine the field
acting on a muon in a disordered intergrain region of the sample.

Let us denote the demagnetization factors of the film as $N_{ik}$
and write the relations between the average fields and the
external field in the form
\begin{equation}\label{n10}
  {\cal B}_i= \langle H_i\rangle +4\pi N_{ik} \langle M_k\rangle,
  \quad \langle {\bf B}\rangle = \langle {\bf H}\rangle
  +4\pi \langle {\bf M}\rangle.
\end{equation}
Let the $z$ axis of the coordinate system associated with the film
be perpendicular to the film plane. Since, by definition, the film
thickness $d$ is much smaller than its linear dimensions $L$, we
have $N_{zz}\equiv N_{\perp} \approx 1$ and all other components
are negligibly small.

If the external field is perpendicular to the film plane
${\boldsymbol{\cal B}}{\|} z$, we have
\begin{equation}\label{n11}
 {\cal B}_z \equiv  {\cal B}=\langle H_z \rangle
  + 4 \pi \langle M_z \rangle, \quad
  \langle H_{\rm pl} \rangle \approx  0,
\end{equation}
where $H_{\rm pl}$ are the vector ${\bf H}$ components lying in
the film plane. From the relation (\ref{n10}) we obtain
\begin{equation}\label{n12}
  \langle B_z \rangle = {\cal B}, \quad
  \langle B_{\rm pl} \rangle \approx 4 \pi
   \langle M_{\rm pl} \rangle.
\end{equation}
If the external field lies in the film plane ${\boldsymbol{\cal
B}}{\perp} z$, we have
\begin{equation}\label{n13}
  \langle H_z \rangle + 4 \pi \langle M_z \rangle=0, \quad
  \langle H_z \rangle =- 4 \pi \langle M_z \rangle, \quad
  \langle H_{\rm pl}\rangle \approx {\cal B}
\end{equation}
and the average induction is given by
\begin{equation}\label{n14}
  \langle B_z \rangle =4\pi \langle M_z \rangle, \quad
  \langle B_{\rm pl} \rangle = {\cal B}+ 4\pi
   \langle M_{\rm pl} \rangle.
\end{equation}

In the accepted model of the film a nanocrystal grain is a
single-domain particle. Therefore, in contrast with the situation
in an ordinary ferromagnet, each grain has no domain wall and is
in both the external (homogeneous) field $\boldsymbol{\cal B}$ and
the field induced by the magnetized grains of the film.

For each grain (in the form of ellipsoid) we can write a relation
similar to Eq. (\ref{n10}), $\langle B_i\rangle= H_i +4\pi
n_{ik}M_k$, where $n_{ik}$ are the grain demagnetization factors.
Analytical expressions could be obtained in the approximation of
oblate ellipsoids, when $n_{\|}=n_{zz} \approx 1, \quad n_{\perp}
=1-n_{\|} \ll 1$.

If the external field is perpendicular to the film plane, Eqs.
(\ref{n12}) give
\begin{equation}\label{n15}
  \langle B_z\rangle\approx {\cal B}, \quad
  \mbox{или} \quad  H_z ={\cal B} - 4\pi M_z \quad
  \mbox{и} \quad
  H_{\rm pl}= \langle B_{\rm pl} \rangle=
  4\pi \langle M_{\rm pl} \rangle, \quad
   B_{\rm pl} =4\pi\bigl(\langle M_{\rm pl} \rangle
   +M_{\rm pl} \bigr).
\end{equation}
If the external field lies in the film plane, we obtain
\begin{equation}\label{n16}
  H_z = -4\pi M_z, \quad
  H_{\rm pl} \approx \langle B_{\rm pl} \rangle=
  {\cal B} +4\pi \langle M_{\rm pl} \rangle \quad
  \mbox{и} \quad B_z =0, \quad
  B_{\rm pl}= {\cal B}+4\pi\bigl(\langle M_{\rm pl} \rangle
  + M_{\rm pl}\bigr).
\end{equation}
Thus, to determine the fields, we should determine the direction
of the magnetization vector ${\bf M}$ in each grain and $\langle
{\bf M} \rangle$ in the film. The local field was obtained in
\cite{Bel07} in the approximation when the ordered grains are
considered as oblate ellipsoids.



\section{Muon spin polarization description}

We can see that the local field acting on a muon (see Eqs.
(\ref{n12}) and (\ref{n16})) could be represented as a sum of two
items, one of them is a constant and the other depends on a
magnetization orientation in a grain. An averaging of a muon spin
polarization overall possible grain orientations leads up to an
effective depolarization in accordance with Eq. (\ref{n9}). In a
case of strong external field (${\cal B}\gg M$) a precession
frequency of a muon spin polarization could be written as a sum of
two items too, $\omega=\omega_0+\omega(\vartheta)$, where
$\vartheta$ is the angle between the grain magnetization and the
external field. At this approach only the transverse component of
a muon spin polarization does not equal to zero, and it can be
approximately written in a form
\begin{equation}\label{n17}
  {\cal P}_{\perp}=\langle \frac{b_z}{b}{\e}^{-{\i}\gamma_{\mu} b
  t}\rangle \approx {\e}^{-{\i}\omega_0t}
  \langle{\e}^{-{\i}\omega(\vartheta)t}\rangle=
  {\e}^{-{\i}(\omega_0+\Delta\omega)t}
  {\e}^{-\sigma^2t^2},
\end{equation}
Here the frequency shift $\Delta\omega$ and the depolarization
rate $\sigma$ are determined by both the grain magnetization value
and the muon interstitial position. Analytical expressions for
these important parameters were obtained in \cite{Bel07} in cases
of rapidly diffusing and nondiffusing muons.

In the case of fast diffusing a muon spin polarization behaviour
in fcc and bcc lattices is practically the same. To within terms
quadratic in $M/{\cal B}$ the precession frequency $\omega_0$ and
its shift $\Delta\omega$ are given by
\begin{equation}\label{n18}
  b_0= {\cal B}-\frac{8\pi}{3}M+
  \frac{1}{2}\left(\frac{4\pi}{3}\right)^2\frac{M^2}{\cal B},\quad
  \Delta\omega=\gamma_{\mu}\frac{1}{63}\frac{2\pi}{3}M
  \left(\frac{M}{\cal B}\right)^2.
\end{equation}
The second moment $\sigma_{\rm diff}^2\propto (M/{\cal B})^2$ is
rather small and it is unlikely that it can be measured in
experiments.

In the case of nondiffusing muons one can obtain the more detail
information. In a bcc lattice of ferromagnet belongs to the ``easy
axis'' type (e.g., Fe) the precession frequency and its shift are determined by
\begin{equation}\label{n19}
     \omega_0^{bcc} = \gamma_{\mu} ({\cal B}-(2\pi+
  \frac{a_{\perp}}{2})M),\quad
  \Delta \omega^{bcc} = -\frac{1}{3}\gamma_{\mu}d\, M \left[1-
  \left(2d - \frac{41}{28}\beta\right)
  \frac{2M}{15\cal B}\right],
\end{equation}
and the second moment
\begin{equation}\label{n20}
     \sigma^2_{\rm nd} = \frac{7}{30}\left(\gamma_{\mu} d\,
   M\right)^2.
\end{equation}
Here $a_{\|}$ and $a_{\perp}$ are the components of the tensor
$a_{ik}$ defining the dipole field,  $d = (a_{\|}-a_{\perp})/2$,
$\beta$ is the magnetic anisotropy parameter.

The behaviour of the polarization of nondiffusing muons in
uniaxial ferromagnets is similar to that presented above for cubic
ferromagnets (\ref{n19})--(\ref{n20})
\begin{align}\label{n21}
     \omega_0^{hcp} = \gamma_{\mu} \left[{\cal B}-\left(
  \frac{8\pi}{3}-\frac{\Delta}{2} \right)
  M\right],\quad &
  \Delta \omega^{hcp} = -\frac{1}{2}\gamma_{\mu}M\Delta
    \left[1-\bigl(\beta+\frac{3}{2}\Delta
   \bigr)\frac{4M}{5\cal B}\right], \nonumber \\
   &\sigma^2_{\rm nd} = \frac{21}{40}\left(\gamma_{\mu}
   M \Delta \right)^2.
\end{align}





\section{Conclusion}

The formulas presented in this report show that the
$\mu$SR-technique gives the opportunity to determine the
magnetization in a grain by measuring the precession frequency and
the depolarization rate and to reveal the existence of muon
diffusion in the crystallographically ordered fraction of a
sample. In the case of nondiffusing muons in a
crystallographically ordered grain the precession frequency
measurement allows one to determine the magnetic anisotropy
constant, which is of great importance in the physics of the
phenomenon under consideration. The measurement of the
depolarization rate of the muon spins precessing at the frequency
corresponding to the average magnetic field of the film allows
determination of the characteristic scale of the magnetic field
inhomogeneity in the disordered intergrain region. The ratio
between the precession amplitudes of the two fractions of muons
allows to determine the ratio of the volumes of the paramagnetic
and ferromagnetic phases of the sample. In addition, the local
fields induced by nuclear magnetic moments can be taken into
account separately using the well known approaches for normal
metals \cite{SB}. We note that all stable isotopes of Fe and Ni,
except $^{57}$Fe (2.21\%) and $^{61}$Ni (1.25\%), have zero spins
(see e.g., \cite{Vons}) and, hence, experiments with these
widespread magnets are preferred.

Last experiments \cite{Kozh}-\cite{Stilp} showed that low-energy
muons (LE-$\mu$SR) give opportunities to study magnetic properties
and inhomogeneity of thin superconducting films. This LE-$\mu$SR
technique has a scale resolution up to 10 nm and can be applied
successfully to solve the fundamental problem of magnetic scale
phase transitions in ferromagnets too. Simultaneous measurements
by macroscopic methods (see e.g., \cite{Char}) would make it
possible to obtain complete information on the physics phase
transitions in nanocrystalline ferromagnetic films.


\ack I would like to acknowledge my colleagues for fruitful discussions.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section*{References}

\begin{thebibliography}{99}
\bibitem{BS}
Belousov Y M and Smilga V P 2000 {\it Physica} B {\bf 289-290} 141
\bibitem{Tepl}
Teplykh A E, Bogdanov S G, Valiev E Z, Pirogov A N,
Dorofeev Y A, Ostroushko A A, Udilov A E, Kazantsev V A, and
Kar'kin A E 2003 {\it Phys. Solid State} {\bf 45} 2328
\bibitem{Olh}
Ol'khovic L P, Khvorov M M, Borisova N M, Golubenko Z V,
Sizova Z I and Shurinova E V 2003 {\it Phys. Solid
State} {\bf 45} 675
\bibitem{Frol}
Frolov G I, Zhigalov V S, Zharkov S M, Pol'skii A I and
Kirgizov V V 2003 {\it Phys. Solid State} {\bf 45} 2303.
\bibitem{SB}
Smilga V P and Belousov Y M 1994 {\it The Muon Method in Science} (New York: Nova
Science)
\bibitem{Bel}
Belousov Y M, Gorelkin V N, Mikaelyan A L, Miloserdin V Y
and Smilga V P 1979 {\it Sov.Phys. Usp.} {\bf 22} 679
\bibitem{A}
Abraham A 1961 {\it The principles of nuclear magnetism} (Oxford:
Clarendon)
\bibitem{Bel07}
Belousov Y M 2007 {\it Phys. Solid State} {\bf 49} 288
\bibitem{Vons}
Vonsovskiy S V 1973 {\it Magnetizm of Microparticles} (Moscow: Nauka)[in Russian]
\bibitem{Kozh}
Kozhevnikov E, Suter A, Fritzsche H, Gladilin V, Volodin A, Moorkens T, Trekels M, Cuppens J, Wojek B M, Prokscha T, Morenzoni E, Nieuwenhuys G J, Van Bael M J, Temst K, Van Haesendonck C and Indekeu J O 3013 {\it Phys.Rev.} B {\bf 87} 10
\bibitem{Char}
Charnukha A, Svitcivic A, Prokscha T, Pr\"{o}pper D, Ocelic N, Suter A, Salman Z, Morenzoni E, Deisenhofer J, Tsurkan V, Loidl A, Keimer B and Boris A V 2012 {\it Phys.Rev.Lett.} {\bf 109}(1):017003.
\bibitem{Hof}
Hofmann A, Salman Z, Mannini M, Amato A, Malavolti L, Morenzoni E, Prokscha T, Sessoli R and Suter A 2012 {\it ACS Nano} {\bf 6}(9):8390-6
\bibitem{Saad}
Saadaoui H, Salman Z, Prokscha T, Suter A, Huhtinen H, Paturi P and Morenzoni E 2013 {\it Phys.Rev.} B {\bf 88} 180501(R)
\bibitem{Stilp}
Stilp E, Suter A, Prokscha T, Morenzoni E, Keller H,  Wojek B M, Luetkens H, Gozar A, Logvenov G and Bozovic I 2013 {\it Phys.Rev.} B {\bf 88} 064419
\end{thebibliography}

\end{document}



\bibitem{ISIS}
K. Tr\"ager, A. Breitr\"uck, M. Diaz Trigo et al., Physica B {\bf
289}-{\bf 290}, 662 (2000).
\bibitem{KEK}
Y. Miyake, K. Shimomura, Y. Matsuda et al., Physica B {\bf
289}-{\bf 290}, 666 (2000).

%\bibitem{LL}
%Л. Д. Ландау, Е. М. Лифшиц.  Электродинамика сплошных сред. Наука,
%М. (1982).
%\bibitem{Seeger}
%A. Seeger.  Hyp. Int. {\bf 17-19}, 75 (1984).
%\bibitem{Schmolz}
%M. Schmolz, K.-P. Doring, K. F\"{u}rderer et al., {\em Hyp.
%Int.\/} {\bf 31}, 199 (1986).
\vspace{3cm}

\noindent Юрий Михайлович Белоусов\\
Московский физико-технический институт,\\
кафедра теоретической физики\\
тел. (095) 408-7590\\
141700, г. Долгопрудный, Московской обл., \\
ул. Циолковского 32/12, кв. 98.\\
(095) 574-7119

\vfill
 \pagebreak

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centering \includegraphics[height=6cm]{axis.eps}
 \caption{}
\label{fig:axis}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
.

 \vfill \pagebreak

Подпись к рис. \ref{fig:axis}

Ориентация кристаллографических осей зерна одноосного кристалла и
направления векторов средней магнитной индукции $\langle \bf
{B}\rangle$ и намагниченности зерна $\bf{M}$ в случаях {\it a} --
внешнее поле перпендикулярно плоскости пленки ${\boldsymbol{\cal
B}}\| z$ и {\it b} -- внешнее поле лежит в плоскости пленки
${\boldsymbol{\cal B}}\perp z$
\end{document}



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